Problem: Solve
\[\arctan \frac{1}{x} + \arctan \frac{1}{x^3} = \frac{\pi}{4}.\]
From the given equation,
\[\tan \left( \arctan \frac{1}{x} + \arctan \frac{1}{x^3} \right) = \tan \frac{\pi}{4} = 1.\]Then from the addition formula for tangent,
\[\frac{\frac{1}{x} + \frac{1}{x^3}}{1 - \frac{1}{x} \cdot \frac{1}{x^3}} = 1,\]or
\[\frac{x^3 + x}{x^4 - 1} = 1.\]Hence, $x^4 - 1 = x^3 + x,$ or $x^4 - x^3  - x - 1 = 0.$  We can factor this as
\begin{align*}
(x^4 - 1) - (x^3 + x) &= (x^2 - 1)(x^2 + 1) - x(x^2 +1) \\
&= (x^2 + 1)(x^2 - x - 1).
\end{align*}The factor $x^2 + 1$ has no real roots, so $x^2 - x - 1 = 0.$  By the quadratic formula,
\[x = \frac{1 \pm \sqrt{5}}{2}.\]If $x = \frac{1 - \sqrt{5}}{2},$ then $x$ is negative, so
\[\arctan \frac{1}{x} + \arctan \frac{1}{x^3}\]is negative.  Therefore, $x = \boxed{\frac{1 + \sqrt{5}}{2}}.$